Mixed convection of a viscous dissipating fluid about a vertical flat plate

Applied Mathematical Modelling 31 (2007) 843–853

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Mixed convection of a viscous dissipating fluidabout a vertical flat plate

Orhan Aydın *, Ahmet Kaya

Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey

Received 1 May 2005; received in revised form 1 November 2005; accepted 21 December 2005Available online 3 March 2006

Abstract

In this study, the effect of the viscous dissipation in steady, laminar mixed convection heat transfer from a heated/cooled vertical flat plate is investigated in both aiding and opposing buoyancy situations. The external flow field is assumedto be uniform. The governing systems of partial differential equations are solved numerically using the finite differencemethod. A parametric study is performed in order to illustrate the interactive influences of the governing parameters,mainly, the Richardson number, Ri (also known as the mixed convection parameter) and the Eckert number, Ec on thevelocity and temperature profiles as well as the friction and heat transfer coefficients. Based on the facts the free streamis either in parallel or reverse to the gravity direction and the plate is heated or cooled, different flow situations are iden-tified. The influence of the viscous dissipation on the heat transfer varied according to the situation. For some limitingcases, the obtained results are validated by comparing with those available from the existing literature. An expression cor-relating Nu in terms of Pr, Ri and Ec is developed.� 2006 Elsevier Inc. All rights reserved.

Keywords: Mixed convection; Vertical plate; Viscous dissipation; Eckert number; Richardson number

1. Introduction

The mixed (combined forced and free) convection arises in many natural and technological processes (see[1,2]). Depending on the forced flow direction, the buoyancy forces may aid (aiding or assisting mixed convec-tion) or oppose (opposing mixed convection) the forced flow, causing an increase or decrease in heat transferrates [3].

The problem of mixed convection resulting from the flow over a heated vertical plate is of considerable the-oretical and practical interest. A detailed review of the subject, including exhaustive lists of references, can befound in the books by Gebhart et al. [2], Bejan [4], Pop and Ingham [5], Jaluria [6] and Chen and Armaly [7].

Refs. [8–13] are some examples of the recent relevant studies existing in the literature. Kafoussias et al. [8]used a modified and improved numerical solution scheme, for local non-similarity boundary layer analysis, to

0307-904X/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2005.12.015

* Corresponding author. Tel.: +90 462 377 2974; fax: +90 462 325 5526.E-mail address: [email protected] (O. Aydın).

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844 O. Aydın, A. Kaya / Applied Mathematical Modelling 31 (2007) 843–853

study the combined free-forced convective laminar boundary layer flow, past a vertical isothermal flat plate,with temperature-dependent viscosity. Hossain and Munir [9] considered a two-dimensional mixed convectionflow of a viscous incompressible fluid of temperature-dependent viscosity past a vertical impermeable fluid.Mulaweh [10] conducted experiments on laminar mixed convection flow adjacent to an inclined heated flatplate with uniform wall heat flux. Merkin and Pop [11] used a similarity transformation to analyze mixed con-vection boundary-layer flow over a vertical semi-infinite plate in which the free stream velocity is uniform andthe wall temperature is inversely proportional to the distance along the plate. Steinruck [12] found a new sim-ilarity solution of mixed convection flow along a vertical plate. In a recent study, Pantokratoras [13] studiedthe steady laminar mixed convection of water with density–temperature relationship along a vertical isother-mal plate.

Despite its importance especially in highly viscous fluids with a low thermal conductivity, the presence ofviscous dissipation has been generally neglected in the existing literature. Israel-Cookey et al. [14] investigatedthe influence of viscous dissipation and radiation on the problem of unsteady magneto-hydrodynamic free-convection flow past an infinite vertical heated plate in an optically thin environment with time-dependent suc-tion. El-Amin [15] investigated the influence of viscous dissipation on buoyancy-induced flow over a horizon-tal or vertical flat plate embedded in a non-Newtonian fluid saturated porous medium under the action oftransverse magnetic field.

The objective of the present paper is to consider mixed convection from a vertical plate in the presence ofviscous dissipation effect. Numerical results are presented for some representative values of governing param-eters, mainly, the Richardson number representing the mixed convection parameter and the Eckert numberrepresenting the effect of the viscous dissipation.

2. Analysis

Consider the steady, laminar, incompressible, two-dimensional, mixed convection boundary-layer flow overa vertical flat plate shown in Fig. 1. The viscous dissipation effect is taken into consideration. The coordinatesystem is chosen such that x measures the distance along the plate and y measures the distance normal to it.Far away from the plate, the velocity and the temperature of the uniform main stream are U1 and T1, respec-tively. The entire surface of the plate is maintained at a uniform temperature of Tw. In the analysis, all thethermophysical properties are assumed to be constant except that the density in the buoyancy term. Assumingthat the Boussinesq approximation is valid, the boundary-layer form of the governing equations which arebased on the balance laws of mass, momentum and energy can be written as

ouoxþ ov

oy¼ 0; ð1Þ

uouoxþ v

ouoy¼ t

o2uoy2� gbðT � T1Þ; ð2Þ

uoToxþ v

oToy¼ a

o2T

oy2þ l

qcp

� �ouoy

� �2

. ð3Þ

Here u and v are the velocity components parallel and perpendicular to the plate, T is the temperature, b is thecoefficient of thermal expansion, l is the dynamic viscosity, t is the kinematic viscosity, q is the fluid density, cp

is the specific heat at constant pressure, g is the acceleration due to gravity, and a is the thermal diffusivity. Theplus and minus signs of the buoyancy term denote the upward and downward flows of free stream,respectively.

The appropriate boundary conditions for the velocity and temperature of this problem are as follows:

x ¼ 0; y > 0; u ¼ U1; T ¼ T1;

x > 0; y ¼ 0; u ¼ 0; v ¼ 0; T ¼ T w ¼ constant;

y large; u! U1; T ! T1.

ð4Þ

Here, U1 and T1 are the free stream velocity and temperature, respectively.

O. Aydın, A. Kaya / Applied Mathematical Modelling 31 (2007) 843–853 845

With definition of the following dimensionless variables:

U ¼ uU1

; V ¼ vRe1=2L

U1; X ¼ x

L; Y ¼ yRe1=2

L

L; h ¼ T � T1

T w � T1. ð5Þ

Eqs. (1)–(3) are converted to the following dimensionless forms:

oUoXþ oV

oY¼ 0; ð6Þ

UoUoXþ V

oUoY¼ o2U

oY 2� GrL

Re2L

; ð7Þ

UohoXþ V

ohoY¼ 1

Pro

2h

oY 2þ Ec

oUoY

� �2

; ð8Þ

where

Pr ¼ lcp

k; Ec ¼ U 2

1cpðT w � T1Þ

; Gr ¼ gbðT � T1ÞL3

t2;

Re ¼ U1Lt

and Ri ¼ Gr

Re2.

ð9Þ

Pr is the Prandtl number, Ec is the Eckert number, Gr is the Grashof number, Re is the Reynolds number, Ri

is the Richardson number.In the dimensionless form, the boundary conditions can be written as follows:

X ¼ 0; Y > 0; U ¼ 1; h ¼ 0;

X > 0; Y ¼ 0; U ¼ V ¼ 0; h ¼ 1;

Y large; U ! 1; h! 0.

ð10Þ

Eqs. (6)–(8) are coupled non-linear differential equations to be solved under the boundary conditions given inEq. (10). However, exact or approximate solutions are not possible for this set of equations and hence we solvethese equations by using the finite-difference method. The equivalent finite difference schemes correspondingEqs. (6)–(8) are given by

Ui�1;jU i;j � U i�1;j

DXþ V i�1;j

U i;jþ1 � U i;j�1

2DY¼ Ui;jþ1 þ U i;j�1 � 2U i;j

ðDY Þ2� Rihi;j; ð11Þ

Ui�1;jhi;j � hi�1;j

DXþ V i�1;j

hi;jþ1 � hi;j�1

2DY¼ 1

Prhi;jþ1 þ hi;j�1 � 2hi;j

ðDY Þ2þ Ec

Ui;jþ1 � U i;j�1

2DY

� �2

. ð12Þ

After some derivations V values can be determined from the continuity equation as

V i;j ¼ V i;j�1 �DY

2DX

� �U i;j � U i�1;j þ U i;j�1 � Ui�1;j�1

� �. ð13Þ

Here the index i refers to x and j to y. The above equations are explicit in x-direction, while they are implicit iny-direction. After specifying the conditions along some initial i = 1, U values and, in the following, h valuescan be obtained on the i = 2 line. Then, V values are obtained on the i = 2 line. Having in this way determinedthe values of all the variables on the i = 2 line, the same procedure can then be used to find the values on thei = 3 line and so on. More detail on the numerical procedure can be found in the textbook by Oosthuizen andNaylor [16]. A mesh system with 100 · 100 nodes is proven to suggest mesh-independent results.

The Nusselt number can be defined as follows:

Nux

Re1=2¼ �X

hw

ohoY

� �Y¼0

. ð14Þ
L


3. Results and discussion

The schematic of the problem examined in this study has been already shown in Fig. 1. The governing equa-tions (6) and (8) subject to the boundary conditions given by Eq. (10) have been solved numerically for thefollowing ranges of the main parameters: Ri = �10, �1, �0.1, �0.01, 0.01, 0.1, 1, 10; Pr > 1; Ec = �0.5–0.5. The combined effects of the Richardson number and the Eckert number on the momentum and heat trans-fer are analyzed and discussed. The Richardson number, Ri, represents a measure of the effect of the buoyancyin comparison with that of the inertia of the external forced or free stream flow on the heat and fluid flow.Outside the mixed convection region, either the pure forced convection or the free convection analysis canbe used to describe accurately the flow or the temperature field. Forced convection is the dominant modeof transport when Ri! 0, whereas free convection is the dominant mode when Ri!1. Buoyancy forcescan enhance the surface heat transfer rate when they assist the forced convection, and vice versa. Viscous dis-sipation, as an energy source, severely distorts the temperature profile. Remember positive values of Ec cor-respond to wall heating (heat is being supplied across the walls into the fluid) case (Tw > Tc), while theopposite is true for negative values of Ec.

First of all, in order to assess the accuracy and validity of our method, we have compared our results for thecase without the effect of the viscous dissipation (Ec = 0) with those given by Saeid [17]. As seen from Table 1,there exists a good correspondence for the results of Nu=Re1=2

L , which gives a credit to the validity of theapproach followed here. As it is clear from this table, increasing the Prandtl number results in the increasein the heat transfer due to the decreasing thermal boundary layer thickness.

Interestingly, according to the direction of the free stream flow and thermal boundary condition applied atthe wall, four different flow situations arise, which are

y, v

x, ug

U∞, T∞

Tw

Fig. 1. The schematic representation of the problem.

Table 1Comparing results for Nu/Re1/2 at Ec = 0

Ri Pr = 0.72 Pr = 7.0

Saeid [17] Present Saeid [17] Present

0.0 0.309 0.298 0.628 0.6250.2 0.332 0.333 0.698 0.6940.4 0.361 0.359 0.752 0.7490.6 0.382 0.381 0.791 0.7880.8 0.402 0.398 0.822 0.8191.0 0.416 0.413 0.851 0.848

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

X

0.0

0.2

0.4

0.6

0.8

1.0

U U

Ri=2, 1, 0

Ec=0.0Pr=10

0.00 0.01 0.02 0.03 0.04 0.05 0.06

X

0.0

0.2

0.4

0.6

0.8

1.0

θ θ

Ri=2, 1, 0

Ec=0.0Pr=10

0.00 0.01 0.02 0.03 0.04 0.05 0.06

X

0.0

0.2

0.4

0.6

0.8

1.0

Ri=2, 1, 0

Ec=0.1Pr=10

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

X

0.0

0.2

0.4

0.6

0.8

1.0

Ri=2, 1, 0

Ec=0.1Pr=10

(a) (b)

Fig. 2. Velocity and temperature profiles for different values of Ri at (a) Ec = 0 and (b) Ec = 0.1 (Case A).

0.30

0.60

0.90

3.00

6.00

9.00

0.09

0.06

0.03

Ri

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.151.20

Ec=0.1, 0.0

Pr=10

Nu/

Re1

/2

Fig. 3. Variation of Nu=Re1=2L for Case A.


0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

X X

0.0

0.2

0.4

0.6

0.8

1.0

U URi=0, - 1, - 2

Ec=0.0Pr=10

0.00 0.01 0.02 0.03 0.04 0.05 0.06

X

0.0

0.2

0.4

0.6

0.8

1.0

θ

θ

Ri=0, - 1, - 2

Ec=0.0Pr=10

0.00 0.01 0.02 0.03 0.04 0.05 0.06

X

0.0

0.2

0.4

0.6

0.8

1.0

Ri=0, - 1, - 2

Ec= - 0.1 Pr=10

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200.0

0.2

0.4

0.6

0.8

1.0

Ri=0, - 1, - 2

Ec= - 0.1Pr=10

(a) (b)

Fig. 4. Velocity and temperature profiles for different values of Ri at (a) Ec = 0 and (b) Ec = �0.1 (Case B).

0.03

0.06

0.09

0.30

0.60

0.90

3.00

6.00

9.00

Ri

0.46

0.48

0.50

0.52

0.54

0.56

0.58

0.60

0.62

0.64

0.66

0.68

0.70

Nu/

Re1/

2

Ec=0.0, - 0.1

Pr=10

Fig. 5. Variation of Nu=Re1=2L for Case B.



1. Case A. Aiding mixed flow with Ec+: The buoyancy force for the wall heating case, Tw > T1 accelerates theupward external forced or free stream flow.

2. Case B. Opposing mixed flow with Ec�: The buoyancy force for the wall cooling case, Tw < T1 retards theupward external forced or free stream flow.

3. Case C. Opposing mixed flow with Ec+: The buoyancy force for the wall heating case, Tw > T1 retards thedownward external forced or free stream flow.

4. Case D. Aiding mixed flow with Ec�: The buoyancy force for the wall cooling case, Tw < T1 accelerates thedownward external forced or free stream flow.

3.1. Case A: Aiding mixed flow with Ec+

For the heated wall case (Tw > T1), the upward free flow caused by the buoyancy is in the same directionwith the external forced flow. This case is called aiding mixed flow. For the case without the viscous dissipa-tion effect (Ec = 0), Fig. 2(a) shows velocity and temperature profiles for different values of the Richardson

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

X

0.0

0.2

0.4

0.6

0.8

1.0

U

Ri=0, - 1, - 2

Ec=0.1Pr=10

0.00 0.01 0.02 0.03 0.04 0.05 0.06

X

0.0

0.2

0.4

0.6

0.8

1.0

θ

Ri=0, - 1, - 2

Ec=0.1Pr=10

Fig. 6. Velocity and temperature profiles for different values of Ri at Ec = 0.1 (Case C).


number, Ri and for different thermal cases at wall. An increase at Ri results in increasing velocities due to addi-tion of buoyancy-induced flow onto the external forced flow. Ec assumes positive values for the heated wallcase. Fig. 2(b) shows the effect of Ri on the velocity and temperature distributions at Ec = 0.1. For the case ofRi = 0, the Eckert number does not have any influence on velocity profile since the momentum and energyequations are not coupled; however, it does on the temperature profile. The viscous dissipation, as a heat gen-eration inside the fluid, increases the bulk fluid temperature. The effect of Ri on the heat transfer from the wallinto the fluid for various Ec is illustrated in Fig. 3. As expected, for the aiding or assisting mixed convection,increasing Ri increases Nu. However, magnitude of this increase decreases with an increase at Ec as a result ofdecreased temperature gradient at the wall, as explained above.

3.2. Case B: Opposing mixed flow with Ec�

For the wall cooling case (Tw < T1), the buoyancy causes a downward free flow which is in the oppositedirection to that of upward external forced flow, which is called the opposing mixed flow. For the case withoutthe effect of the viscous dissipation, Fig. 4(a) shows velocity and temperature profiles for different values of theRichardson number, Ri. An increase at Ri in negative direction results in decreasing velocities due to retardingeffect of downward buoyancy-induced flow onto the upward external forced flow. For the cooled wall case, Ec

receives negative values. For Ec = �0.1, Fig. 4(b) displays the effect of Ri on the velocity and temperature dis-tributions. In this case, for the negative value of Ec, since Tw < T1, the viscous dissipation will again increasethe temperature distribution in the flow region. Finally, this leads to an increased temperature gradient, as willbe shown later, which will result in increased heat transfer values. For this opposing mixed convection case,the effect of Ri on the heat transfer from the wall into the fluid for various Ec is illustrated in Fig. 5. Asexpected, increasing Ri in the negative direction decreases Nu. However, an increase at Ec in the negativedirection as a result of increased temperature gradient at the wall increases Nu.

3.3. Case C: Opposing mixed flow with Ec+

In Cases A and B, an upward forced flow is assumed, while Cases C and D assuming a downward one. Theupward flow caused by the buoyancy for the heated wall case (Tw > T1) has a retarding effect on the externalforced flow, which is called as opposing mixed flow. For the case without the viscous dissipation effect(Ec = 0), the velocity and temperature profiles for different values of the Richardson number, Ri, are symmet-rically identical to those given above for the Case B. An increase at Ri results in decreasing velocities because

0.03

0.06

0.09

0.30

0.60

0.90

3.00

6.00

9.00

Ri

0.46

0.48

0.50

0.52

0.54

0.56

0.58

0.60

0.62

0.64

0.66

0.68

Nu/

Re1/

2

Ec=0.1, 0.0

Pr=10

Fig. 7. Variation of Nu=Re1=2L for Case C.


of opposing effect of the upward buoyancy-induced flow on the downward external forced flow. Including theviscous dissipation effect (Ec = 0.1), the effect of Ri on the velocity and temperature distributions is shown inFig. 6. The effect of Ri on the heat transfer from the wall to the fluid for various Ec is illustrated in Fig. 7. Theincreasing viscous dissipation decreases the temperature gradient near the wall by increasing fluid temperaturefor the wall heating case (Ec > 0). Then, in addition to opposing effect of the buoyancy, viscous dissipation willhave an opposing effect on the heat transfer, too.

3.4. Case D: Aiding mixed flow with Ec�

The downward flow caused by the buoyancy for the cooled wall case (Tw < T1) will aid the downwardexternal forced flow, which is called as aiding or assisting mixed flow. The case without the viscous dissipationeffect (Ec = 0) represents velocity and temperature profiles symmetrically identical to those given above for theCase A. An increase at Ri increases velocities because of aiding effect of the downward buoyancy-induced flowon the downward external forced flow. Including the viscous dissipation effect (Ec = �0.1), the effect of Ri onthe velocity and temperature distributions is shown in Fig. 8. The effect of Ri on the heat transfer from the

0.00 0.01 0.02 0.03 0.04 0.05 0.06

X

0.0

0.2

0.4

0.6

0.8

1.0

θ

Ri= 2, 1, 0

Ec= - 0.1Pr=10

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

X

0.0

0.2

0.4

0.6

0.8

1.0

U

Ri=2, 1, 0

Ec= - 0.1Pr=10

Fig. 8. Velocity and temperature profiles for different values of Ri at Ec = �0.1 (Case D).

0.03

0.06

0.09

0.30

0.60

0.90

3.00

6.00

9.00

Ri

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Nu/

Re1/

2

Ec=0.0, - 0.1

Pr=10

Fig. 9. Variation of Nu=Re1=2L for Case D.

Table 2Values of the coefficients seen in Eq. (15)

Coefficient Case A Case B Case C Case D

a 0.398 0.294 0.294 0.397b 0.111 0.016 0.0217 0.128c 0.173 �0.074 0.067 �0.171R 0.997 0.988 0.987 0.995


fluid to the wall (the wall cooling case) for various Ec is illustrated in Fig. 9. The increasing viscous dissipationincreases the temperature gradient near the wall by increasing fluid temperature for the wall cooling case(Ec < 0). In the following, in addition to aiding effect of the buoyancy, viscous dissipation will have an aidingeffect on the heat transfer, too.

Finally, for the practical use, an expression correlation Nu in terms of Pr, Ri and Ec is developed. Note thatthe viscous dissipation becomes considerable for highly viscous fluids, for which Pr > > 1. Therefore, in addi-tion to the values of Pr studied above, Pr = 1, the results are also obtained for Pr = 10. Above, we only coverthe range of Ec, �0.1–0.1. For deriving such a Nu correlation, this range of Ec is extended to �0.5–0.5. Afterobtaining Nu values for the above values of Pr, Ri and Ec, the following correlation is obtained:

Nu

Re1=2¼ aðRiÞbðPrÞ1=4ð1� cEcÞ; ð15Þ

where R is the correlation coefficient representing the degree of the harmony. The values of the coefficients aregiven in Table 2.

4. Conclusions

Mixed convection flow about a vertical flat plate considering the effect of viscous dissipation is analyzed.Uniform suction/injection was allowed at the wall. Depending on the thermal boundary conditions appliedat the wall (heated/cooled wall) and the direction of forced or free stream flow (upward/downward), four dif-ferent mixed convection flow situations have been identified:

(i) aiding buoyancy with opposing viscous dissipation,(ii) opposing buoyancy with aiding viscous dissipation,


(iii) opposing buoyancy with opposing viscous dissipation,(iv) aiding buoyancy with aiding viscous dissipation.

Finally, a Nu = f(Pr,Ri,Ec) correlation which is valid for all the cases described above is suggested.

References

[1] Y. Jaluria, Natural Convection Heat and Mass Transfer, Pergamon Press, Oxford, 1980.[2] B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia, Buoyancy-induced Flows and Transport, Hemisphere, 1988.[3] O. Aydın, Aiding and opposing mechanisms of mixed convection in a shear- and buoyancy-driven cavity, Int. Commun. Heat Mass

Transfer 26 (7) (1999) 1019–1028.[4] A. Bejan, Convection Heat Transfer, Wiley, New York, 1995.[5] I. Pop, D.B. Ingham, Convective Heat Transfer, Pergamon, Amsterdam, 2001.[6] Y. Jaluria, Basic of natural convection, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat

Transfer, Wiley, New York, 1987.[7] T.S. Chen, B.F. Armaly, Mixed convection in external flow, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase

Convective Heat Transfer, Wiley, New York, 1987.[8] N.G. Kafoussias, D.A.S. Rees, J.E. Daskalakis, Numerical study of the combined free-forced convective laminar boundary layer flow

past a vertical isothermal flat plate with temperature-dependent viscosity, Acta Mech. 127 (1–4) (1998) 39–50.[9] M.A. Hossain, M.S. Munir, Mixed convection flow from a vertical flat plate with temperature dependent viscosity, Int. J. Therm. Sci.

39 (2) (2000) 173–183.[10] H.I. Abu-Mulaweh, Measurements of laminar mixed convection flow adjacent to an inclined surface with uniform wall heat flux, Int.

J. Therm. Sci. 42 (1) (2003) 57–62.[11] J.H. Merkin, I. Pop, Mixed convection along a vertical surface: similarity solutions for uniform flow, Fluid Dyn. Res. 30 (2002) 233–

250.[12] H. Steinruck, About the physical relevance of similarity solutions of the boundary-layer flow equations describing mixed convection

flow along a vertical plate, Fluid Dyn. Res. 32 (1–2) (2003) 1–13.[13] A. Pantokratoras, Laminar assisting and mixed convection heat transfer from a vertical isothermal plate to water with variable

physical properties, Heat Mass Transfer 40 (2004) 581–585.[14] C. Israel-Cookey, A. Ogulu, V.B. Omubo-Pepple, Influence of viscous dissipation and radiation on unsteady MHD free-convection

flow past an infinite heated vertical plate in a porous medium with time-dependent suction, Int. J. Heat Mass Transfer 46 (2003) 2305–2311.

[15] M.F. El-Amin, Combined effect of magnetic field and viscous dissipation on a power-law fluid over plate with variable surface heatflux embedded in a porous medium, J. Magn. Magn. Mater. 261 (1–2) (2003) 228–237.

[16] P.H. Oosthuizen, D. Naylor, Introduction to Convective Heat Transfer Analysis, McGraw-Hill, New York, 1999.[17] N.W. Saeid, Mixed convection flow along a vertical plate subjected to time-periodic surface temperature oscillations, Int. J. Therm.

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Mixed convection of a viscous dissipating fluid about a vertical flat plate